McKay correspondence by Miles Reid

Author:Miles Reid
Language: eng
Format: epub
Published: 1997-02-27T16:00:00+00:00

Aj = (z, 2i, r-3i) for i = 0,..., 2k

(0,0,r)

Figure 7: G-Hilb for ^(1,2,-3). Bi is joined to A2i^2, Mi-\, Mi

to Figure |[ How do I know to join (8, 3, 2)—(2,4, 7) by a cone cr, rather than (7,1, 5)—(3, 6,4)? By calculating 2x2 minors of (247)1 we see that the parameter on the corresponding line Ecr G Y should be the ratio xz^ : y"*, where

xz

, y'^ e L(8). The Newton polygon of L(8) is

S fi 4 2 2 S 4

X X y X y X y y

(2,4,7)

xz

(8,3,2)

y2^3

(The figure is not planar: xz"^ and y'^ are "lower".) Here (2,4,7) and (8,3,2) have minima on the two planes as indicated, with common minima on xz^ and y^, so that the linear system \xz'^ : j/^| can be free on L^. But (7,1,5) and (3,6,4) don't have a common minimimum here: (7,1,5) prefers y"^ only, and (3,6,4) prefers xz^ only. If I join (7,1, 5)—(3, 6,4), the linear system \xz'^ : j/^| would have that line as base locus.

The resolution is as in Figure pi The McKay correspondence marks each

exceptional stratum: a line L parametrised by a ratio x"

is marked by

the common character space of a;"

In other words, a linear system such

(13,0,0),

(0,13,0)

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.